PHYSICAL REVIEW APPLIED 12, 014016 (2019)

PHYSICAL REVIEW APPLIED 12, 014016 (2019)Editors’ Suggestion Featured in Physics

Holograms to Focus Arbitrary Ultrasonic Fields through the Skull

Sergio Jiménez-Gambín, Noé Jiménez,∗ José María Benlloch, and Francisco CamarenaInstituto de Instrumentación para Imagen Molecular, Consejo Superior de Investigaciones Científicas, Universitat

Politècnica de València, CAMINO DE VERA s/n, 46022 València, Spain

(Received 22 February 2019; revised manuscript received 6 May 2019; published 10 July 2019)

We report three-dimensional (3D)-printed acoustic holographic lenses for the formation of ultrasonicfields of complex spatial distribution inside the skull. Using holographic lenses, we experimentally, numer-ically, and theoretically produce acoustic beams whose spatial distribution matches target structures of thecentral nervous system. In particular, we produce three types of targets of increasing complexity. First, aset of points are selected at the center of both right and left human hippocampi. Experiments using a skullphantom and 3D-printed acoustic holographic lenses show that the corresponding bifocal lens simulta-neously focuses acoustic energy at the target foci, with good agreement between theory and simulations.Second, an arbitrary curve is set as the target inside the skull phantom. Using time-reversal methods, theholographic beam bends following the target path, in a similar way as self-bending beams do in free space.Finally, the right human hippocampus is selected as a target volume. The focus of the corresponding holo-graphic lens overlaps with the target volume in excellent agreement between theory in free media, andexperiments and simulations including the skull phantom. The precise control of focused ultrasound intothe central nervous system is mainly limited due to the strong phase aberrations produced by refractionand attenuation of the skull. Using the present method, the ultrasonic beam can be focused not only ata single point but overlapping one or various target structures simultaneously using low-cost 3D-printedacoustic holographic lens. The results open alternative paths to spread incoming biomedical ultrasoundapplications, including blood-brain-barrier opening or neuromodulation.

DOI: 10.1103/PhysRevApplied.12.014016

I. INTRODUCTION

Holographic plates are surfaces that when illuminatedby a wave, typically light, modify the phase of the trans-mitted or reflected wavefront in such a manner that acomplex image can be formed [1–3]. In recent years, sub-wavelength thickness holographic metasurfaces have beendesigned using structured materials with subwavelengthresonances, namely metamaterials [4,5]. Analogously, inacoustics, a broad range of locally resonant structures havebeen proposed to obtain a precise control of the wave-front at a subwavelength scale [6,7], including effectivenegative-mass density [8] and/or bulk modulus metama-terials [9,10]. Acoustic metamaterials allow an accuratecontrol of the reflected [11–15] or transmitted wavefronts[16–18]. The use of these structures has been exploited todesign negative-refraction superlenses [19] or hyperbolicdispersion-relation hyperlenses [20] that exhibit subwave-length focusing properties in the near field. Holographiclenses have also been reported in acoustics to generatecomplex acoustic fields [21–24]. Multifrequency holo-grams have also been reported [25]. Equivalently, using

∗[email protected]

phased-array sources, the generation of complex beam pat-terns [26], self-bending and bottle beams [27], or vortexbeams for particle levitation and manipulation has beenreported [28]. Mixed approaches between metamaterialsand phased arrays have also been presented [29].

In these applications, holographic lenses have demon-strated the ability to manipulate acoustic waves in freemedia, i.e., without inhomogeneities. However, whenusing ultrasound in biomedical-engineering applications,in their path acoustic beams encounter multiple tissue lay-ers of complex geometry with nonhomogeneous proper-ties. For instance, an accurate control of the focused beamis at the basis of focused ultrasound therapy techniques,e.g., as in high-intensity focused ultrasound hyperthermia,thermal ablation or histotripsy, or in extracorporeal shock-wave lithotripsy [30,31]. Focusing directly into human softtissues can efficiently be achieved by using conventionalsystems as ultrasound beam aberrations are typically smallin these media [32]. However, when the target tissue laysbehind high-impedance tissues, e.g., soft tissue surroundedby bones, the beam experiences strong aberrations due torefraction, reflection, and absorption processes [33]. Someapplications make use of existing acoustic windows by tar-geting tissues from specific locations. Nevertheless, in thecase of transcranial propagation, skull bones are always

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present in the path towards the central nervous system(CNS). In this way, the precise control of acoustic focusinto the CNS is mainly limited due to the strong phaseaberrations produced by the refraction and attenuation ofthe skull [34].

To overcome these limitations, minimally invasive tech-niques have been developed in the past to design activefocusing systems using the time-reversal (TR) invarianceof the acoustic propagation [35] or phase conjugationmethods [36]. In minimally invasive techniques, a smallacoustic source is introduced into the skull, together witha biopsy catheter. When the catheter reaches the tar-get tissue it radiates a short ultrasonic pulse that travelsoutwards and it is recorded by a hemispherical mul-tielement array surrounding the patient’s head. Then,the elements of the phased array are set to re-emit thetime-reversed recorded waveforms (or phase-conjugatedharmonic signals). Due to spatial reciprocity and time-reversal invariance of the acoustic media, the generatedwavefront focuses at the catheter location, i.e., at targettissue [35]. Later, it was demonstrated that noninvasiveversions of these techniques can be obtained using numeri-cal simulations [37,38]. In these techniques, a tomographicimage is previously obtained from a patient’s head toextract the geometry of the skull and its acoustic proper-ties [38]. Using full-wave simulations, the time-reversedwavefront is calculated by exciting the simulation witha virtual source at the desired focal spot. Then, a phys-ical hemispherical phased array is excited with the syn-thetic time-reversed waveforms [39] or phase-conjugatedsignals [37], and sharp focusing through skull-aberrationlayers is retrieved. Other techniques include the optimiza-tion of phase arrays using magnetic resonance imaging(MRI) to maximize acoustic-radiation-force-induced dis-placements into the target focus [40]. However, up-to-datephased-array systems are restricted to a limited numberof channels, e.g., 1024 for the Exablate® Model 4000(InSightec, Ltd) [41], that can be insufficient to fully recordthe required holographic information in order to conform acomplex beam pattern. In addition, phased arrays are effec-tive, but due to its high economical cost it is desirable touse passive structures to control acoustic beams.

Only a few theoretical works have tackled the problemof beam focusing through aberrating layers using meta-materials [42] or phase plates [43,44]. In Ref. [42] atwo-dimensional (2D) configuration was proposed theoret-ically using a metasurface based on membranes. Recently,the use of phase plates to generate simple focused sourceshave been reported to avoid beam aberrations in transcra-nial propagation [43]. However, the technique was limitedto focus the beam into a single focal spot at the nearfield of the source. Besides, in some nonthermal transcra-nial ultrasound applications such as blood-brain-barrieropening [45] or neuromodulation [46], the ultrasoundbeam might be set to fully cover a geometrically complex

CNS structure rather than focusing over a small focalspot.

In this work, we propose the use of 3D-printed holo-graphic phase plates to produce ultrasonic fields of arbi-trary shape into the human brain. The holographic lensesdesigned in this work allow the reconstruction of complexdiffraction-limited acoustic images. including the compen-sation of the aberrations produced by a skull phantom.

In particular, we theoretically, numerically, and exper-imentally demonstrate the generation of several holo-graphic patterns, of increasing complexity, all with directpractical application to biomedical ultrasound: an arbitraryset of points, an arbitrary curved line, and an arbitrary vol-ume. First, we provide the conditions to generate a simpleholographic pattern, i.e., a set of diffraction-limited focalpoints, as sketched in Fig. 1(b). In particular, we extendthe use of holographic lenses to generate bifocal beams,matching both foci simultaneously to the location of theleft and right human hippocampi. Second, we demonstrate

Source

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FIG. 1. (a) Scheme of the holographic lens focusing over atarget CNS structure. (b) Focusing on a set of arbitrary points(bifocal holographic lens). (c) Focusing over an arbitrary line(self-bending holographic lens). (d) Focusing over an arbitraryvolume (volumetric holographic lens).

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that ultrasonic beams with a curved trajectory along theinternal CNS tissues can also be produced, as Fig. 1(c)shows. In this way, the acoustic beam can be bent follow-ing arbitrary paths producing a self-bending beam insidethe CNS. Finally, we report the generation of a beampattern that overlaps with the volume of a specific CNSstructure, as shown in Fig. 1(d); in particular, we target theright human hippocampus.

II. METHODS

The process of hologram generation is composed offour steps. First, we extract the geometry and acousticproperties of a human skull from x-ray CT images, asshown in Fig. 2(a), and from MRI tomographic imageswe identify the target tissue structure, e.g., the righthuman hippocampus, as shown Fig. 2(b). Second, a back-propagation method is used to calculate the acousticwavefront generated from a set of virtual sources andimpinging on a holographic surface located outside theskull phantom, as shown in Fig. 2(b). Third, the phase-plate lens is generated by using the phase and amplitudeof the recorded wavefront at the holographic surface, asshown in Fig. 2(c). Finally, the lens is excited with a flatand uniform ultrasonic transducer and the target acous-tic image is reconstructed by either theoretical, numericalforward-propagation or experimental methods, as shown inFig. 2(d).

A. Tomographic image acquisition

First, in order to model the skull geometry, we use theCT datasets of a female human head with an isotropicresolution of 1 mm (interpolated to 0.22 mm for the numer-ical simulation) from the National Library of Medicine’sVisible Human Project available for general use by theUniversity of Iowa. Experiments are conducted in a 3D-printed skull phantom, while, in addition, we include full-wave simulations using the acoustical properties of theskull bones. Thus, for the skull-phantom simulations weuse homogeneous acoustical parameters matching those

of the 3D printing material, while for the realistic skullsimulations we use the same geometry but the inhomo-geneous acoustical parameters of the skull are derivedusing the same CT data, converting the apparent densitytomographic data in Hounsfield units to density and sound-speed distributions using the linear-piecewise polynomialsproposed in Refs. [47] and [48].

After, we use a human atlas made publicly available bythe International Consortium for Brain Mapping (ICBM)from the Laboratory of Neuro Imaging [49]. This atlasprovides us T1-weighted MRI data that are used to iden-tify the shape and location of the human hippocampus.In particular, for segmentation, we use ITK-SNAP software[50] to obtain the shape and location of the left and righthippocampi.

B. Calculation methods

We use two methods, of increasing complexity, toestimate the back-and-forward acoustic fields: a semi-analytical method using the Rayleigh-Sommerfeld diffrac-tion integral and a pseudospectral time-domain simulationmethod.

On the one hand, for theoretical calculations in homo-geneous media, i.e., in water without the skull phantom,the acoustic pressure field given by p(r) at point r, gen-erated by a moving surface S of arbitrary shape locatedat coordinates r0 and vibrating with a complex particlevelocity v0(r0) normal to the surface, is given by theRayleigh-Sommerfeld diffraction integral [51]:

p(r, ω) = iωρ0

2π

∫S

v0(r0) exp (−ik0 |r − r0|)|r − r0| dS, (1)

where ω = 2π f ; k0 = ω/c0, c0, and ρ0 are the wave num-ber, sound speed and density of water. Note that in Eq.(1), diffraction is captured exactly as compared with angu-lar spectrum methods, so it can be applied to high-aperturesources.

On the other hand, for calculations including aberrationlayers we use a pseudospectral simulation method with

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FIG. 2. Hologram generation process. (a) CT+MRI tomographic images. (b) Selected target (red volume) acting as a virtual acousticsource and holographic recording surface (blue area). (c) Lens design using the TR back-propagated field. (d) Forward propagationfrom the holographic lens (red area) to the target tissue (blue volume).

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k-space dispersion correction to numerically integrate thelinearized constitutive relations of acoustics [52,53]. In aninhomogeneous and absorbing media, the governing equa-tions, i.e., the continuity equation, the momentum con-servation equation and the pressure-density relation, canbe written as three-coupled first-order partial differentialequations as

∂ρ

∂t= −ρ0∇ · u − u · ∇ρ0, (2)

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= − 1ρ0

∇p , (3)

p = c20 (ρ + d · ∇ρ0 − Lρ) , (4)

where u is the acoustic particle velocity, d is the acous-tic particle displacement, p is the acoustic pressure, ρ isthe acoustic density, ρ0 is the ambient (or equilibrium)density, c0 is the sound speed, and L is a linear operatorintroducing the frequency-dependent absorption and dis-persion [52]. Tissue absorption following a power law onfrequency given by α(ω) = α0ω

γ , where α0 is the absorp-tion coefficient and γ is the exponent of the frequencypower law, together with its corresponding physical dis-persion are included by the integro-differential operatoras

L = τ∂

∂t(−∇2)(γ /2)−1 + η

(−∇2)[(γ+1)/2]−1, (5)

where τ = −2α0cγ−10 and η = 2α0cγ

0 tan (πγ /2) are theabsorption and dispersion proportionality coefficients. Thisoperator is solved efficiently using the fractional Lapla-cian in the k space. This simulation method is selectedas it provides low numerical dispersion as compared withfinite-difference methods [54]. We use a numerical gridwith a spatial step of x = y = z = 223 μm and anumerical temporal step of t = 19.1 ns, leading to aCourant-Friedrichs-Lewy number [52] of 0.13 in waterand a spatial sampling of six grid points per wavelengthin water for a frequency of 1 MHz. These parameters arefixed for all simulations in this paper.

C. Lens design

Under the assumption of reciprocity, time invariance,and linearity of the system, a time-reversal techniquetogether with a direct method is used to design the phase-only holographic lens.

First, we set some virtual sources inside the skull phan-tom and the back-propagated field is estimated at a givensurface outside the skull phantom. For the bifocal lens,two virtual sources are set as monopoles with the samephase and amplitude, located at the center of mass of thetwo hippocampi (right and left), as sketched in Fig. 1(a).For the self-bending beam, a set of 50 virtual sources

are located following an arbitrary curve as sketched inFig. 1(b), each source compensated by a phase factorof exp(ikzz) accounting for the direction of arrival ofthe wavefront. Finally, for the volumetric hologram, assketched in Fig. 1(c), a set of virtual sources are spatiallydistributed with a separation of λ/6 (to match the numer-ical grid used) over a sagittal plane of the right humanhippocampus. The recorded field is captured at a givensurface, i.e., at a holographic surface, outside the skullphantom.

Second, the recorded conjugated pressure distribution atthe working frequency is used to design the physical lens.The lens surface is divided in squared pixels of differentheight, h(x, y) and uniform width, h, as shown in Fig. 3.We assume each elastic column to vibrate longitudinally asa Fabry-Perot resonator. For each column, the field at theholographic plane located at x0 = (x, y, d) is given by thecomplex transmission coefficient [55]:

T(x0) = 2Ze−ik0[d−h(x0)]

2Z cos [kLh(x0)] + i(Z2 + 1

)sin [kLh(x0)]

, (6)

where d is the distance from the bottom of the lens (z = 0)to the holographic surface, the normalized impedance isgiven by Z = ZL/Z0, and Z0 = ρ0c0 is the impedanceof water and ZL = ρLcL, kL = ω/cL, ρL and cL, are theimpedance, wave number, density, and sound speed of thelens material. In order to obtain the height of each pixel ofthe lens, an analytic inversion of Eq. (6) is not possible dueto the trigonometric terms. Instead, we first numericallyevaluate the expression for a broad range of pixel heightsranging from a minimum height that is set to hmin = 5 mmto guarantee structural consistency, to a given height thatprovides a phase of the transmission coefficient 2π greaterthan for hmin, i.e., d = 15 mm, and using steps of 1 μm,well below the printer accuracy. Finally, we perform inter-polation using a cubic-spline method to obtain the height ofthe pixel as a function of the required phase. In this way, bytuning the height of each Fabry-Perot resonator the phaseat the output of each pixel can be tailored to that of a targetholographic surface.

FIG. 3. Geometry of the holographic lens. The lens, of aperture2a is subdivided in pixels of height h(x, y). The source is locatedat z = 0, while the holographic plane is located at z = d.

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However, using this kind of lens the degree of freedomto modify the magnitude of the field at the holographicsurface is limited. Iterative methods have been employedin the past to obtain equivalent lenses only with phasedistributions [21]. In this work, iterative methods are pro-hibitive: the 3D simulations including aberration layersinvolve long calculation times, e.g., 20 h in a Intel® Xeon®

CPU E5-2680 v2 2.80 GHz, 256 GB RAM, using a CPUparallel implementation of the code. Instead, we use adirect method to estimate an equivalent holographic lensof uniform field magnitude [56]. The basis of this directmethod is the sequential scanning of the pixels to modifythe complex transmission coefficient. The method worksas follows: first, the odd and even rows are scanned fromopposite directions, and a bidirectional error of the diffu-sion process is calculated. The magnitude of each visitedpixel is forced to be a constant value while the exactphase value is preserved. The resulting error is diffusedto the neighboring pixels. Finally, the result gives a sur-face with a modified phase depending on the bidirectionalerror-diffusion process [56]. The main limitation of thismethod is that if the pixel width is small it appears in areaswith isolated long pixels, i.e., columns, that can experi-ence bending modes. Note this does not imply that a lenscannot be designed, but the theory we present here onlyapplies to longitudinal modes on each pixel. The size ofthe pixels used in this work, 5/6 times the wavelength, isthick enough to ensure that the resonance frequency of thefirst bending mode is far away from the first longitudinalFabry-Perot resonance frequency.

Third, lenses with an aperture of 2a = 50 mm are man-ufactured using 3D printing techniques. On the one hand,the bifocal holographic lens is manufactured by additive3D printing techniques using Ultimaker 3 Extended (Ulti-maker B.V., The Netherlands) with a resolution of 100 μmin both lateral and axial directions and PLA material. As, ingeneral, the height profile of the lens is smooth, we set thesquare pixel resolution to h = 0.22 mm. The acousticalproperties of polylactic acid material are obtained exper-imentally using a pulse-echo technique in a test cylinder,resulting in a measured sound speed of cL = 1818 m/sand a density of ρL = 1127 kg/m3, matching with thosereported in the existing literature [57], and the absorptionis set to α = 13.72 dB/cm at 1.112 MHz [57]. On theother hand, the self-bending and volumetric holographiclenses, which needed a more accurate printing techniquefor their complex pattern, are 3D printed using Polyjettechniques with an Objet30 printer (Stratasys, USA), witha resolution of 100 μm and 28 μm in lateral and axialdirections, respectively, and using a photoresistive poly-mer (Veroclear®, Stratasys, USA). As a result of the directmethod to obtain the equivalent holographic lens of uni-form field magnitude [56], the height distribution presentshigh spatial modulations [see Fig. 2(c)]. Thus, the pixelresolution is increased to h = 1 mm to ensure each

column vibrates as a longitudinal Fabry-Perot resonatoravoiding bending modes around the working frequencyfor the volumetric hologram. For this material we exper-imentally estimate cL = 2312 m/s and ρL = 1191 kg/m3

and α = 3.06 dB/cm at 1.112 MHz, matching the valuesreported in the existing literature [21].

D. Skull phantom

The geometry of the skull phantom is extracted fromthe 3D CT images as described previously. The soundspeed and density distributions are first estimated from theapparent density given by the CT images in Hounsfieldunits [47,48]. Then, as the 3D printing technique results inhomogeneous material, the acoustic properties, includingthe absorption are considered uniform along the skull-bone volume [58–60]. The skull phantom is manufacturedby additive 3D printing techniques using Ultimaker 3Extended (Ultimaker B.V., The Netherlands) with res-olution of 100 μm in both lateral and axial directionsand using PLA material. The acoustic parameters for the3D-printed phantom are the same as for the PLA lenses.

Finally, for the simulations using a realistic skull, theacoustical parameters are derived using the CT data, con-verting the apparent density in Hounsfield units to densityand sound-speed distributions using the linear-piecewisepolynomials proposed in Refs. [47] and [48]. The den-sity data ranges between ρ0 = 1000 kg/m3 (water) andρmax = 2206 kg/m3 (bone), the sound-speed values rangebetween c0 = 1500 m/s and cmax = 3117 m/s, match-ing those reported in the literature [31,61] and the boneabsorption is set to 12.6 dB/cm at 1.112 MHz [62].

Details about the measurement system can be found inthe Appendix.

III. MULTIPLE-POINT HOLOGRAMS

We start with the bifocal holographic lens. First, twopoints located at the center of mass of both left and righthuman hippocampi are selected. Second, we set this pair ofpoints as the location of virtual sources for the TR method.For the lens designs in free media, i.e., without the skull,we make use of the Rayleigh-Sommerfeld diffraction inte-gral (see Sec. II for further details). For the lens designsof holographic surfaces including the skull-aberration lay-ers we make use of low-numerical-dispersion simulationsbased on pseudospectral methods [52]. In this way, thesimultaneous back propagation of the fields irradiated byboth virtual monopoles can be calculated at the holo-graphic surface, which is located at the rear part of theskull. The phase-plate lens is designed using the conju-gated complex field recorded at the surface. Then, thelens is placed at the location of the holographic surfaceas shown in Fig. 4(a), and a forward-propagation calcula-tion is carried out to test the quality of the reconstructedacoustic image.

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(a)(c)

(d)

(e)

(f)

(b)

FIG. 4. Axial-field cross section obtained for the bifocal lens using simulations (a) and experiments (b). (c),(d) Correspond-ing transversal-pressure-field distributions. Color bar in units normalized to the peak pressure. (e),(f) Simulated and experimentalnormalized axial- and transversal-field cross sections, respectively.

The field produced by the bifocal lens propagatingthrough a human occipital-parietal skull phantom, includ-ing the compensation for the aberrations of the skullis shown in Figs. 4(a)–4(f). First, Figs. 4(a) and 4(b)show the axial-field cross section, p(x, y = 0, z), using thepseudospectral simulation method and measured experi-mentally, respectively. We observe that the reconstructedfield accurately matches the target foci, and the exper-imental results agree with the simulations. The corre-sponding transverse-field distributions at z = 105 mm areshown in Figs. 4(c) and 4(d) where sharp focusing isobserved. The focal spots present larger dimensions in theaxial (z) direction than in transverse ones, as expectedfrom limited-aperture holographic lenses, where the spa-tial spectral components in axial direction are limited bythe finite-aperture source [63]. Axial (measured at x =25 mm and y = 0 mm) and transversal (measured atz = 105 mm and y = 0 mm) cross sections are shownin Figs. 4(e) and 4(f), respectively. Excellent agreementis observed between simulation and experiment for theaxial-field profile at the focal region. A small secondarylobe located before the main one appears experimen-tally. The transverse profile shows a small lateral shift of±0.5 mm in both experimental foci towards the x-axisorigin.

Note that, due to diffraction, the geometrical focus of ageometrically focused source does not correspond to theacoustic focus of the source [63]. In our case, the targetlocation is set to z = 105.5 mm, the acoustical focus ofan equivalent focused source of the same frequency and

aperture in water peaks at z = 99.8 mm and the focusof the lens peak at z = 100.4 mm and z = 100.1 mm insimulations and experiments including the skull phantom,respectively. These shifts correspond to errors of 0.6 and0.3%, respectively, showing the accuracy of the focusingperformance of the holographic lenses.

IV. SELF-BENDING BEAMS

The previous results show that holographic phase platescan retain phase information of multiple foci. Using thisidea, we can set more complex targets following the shapeof functional structures found in the CNS. Here, we set thetarget holographic pattern to a beam following a curvedtrajectory as those reported in homogeneous media with-out aberrating layers using active sources or metamate-rials [17,27,64]. As the aberration layers are present inthe real application, known analytical methods to calcu-late the phase of the 3D trajectory are, in principle, notavailable [27]. Instead, we make use of a TR method:a set of virtual sources are placed along this trajectoryand their back-propagated field are calculated. We set afactor of (z/zmax) exp(izkz) to compensate for the ampli-tude and phase of each source to set the main directionof propagation, being kz the axial component of the wave-vector and zmax the distance to the farthest virtual source(45 mm in this example). A sketch of the target tra-jectory is shown in Fig. 5(a). The axial and transversalcross sections of the forward-propagated field in water areshown in Figs. 5(a) and 5(b), respectively. We observe

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H L L

T

T T T

(a)

(b)

(c) (e)

(d) (f)

FIG. 5. Theoretical (a) axial- and (b) transverse-pressure-field distribution for the self-bending beam in water. Simulated (c) axial-and (d) transverse-pressure-field distribution for the self-bending beam in water. Corresponding experimental results are shown in theinsets in (c) and (d). (e),(f) Simulated axial and transversal pressure field, including the skull phantom. Corresponding experimentalresults are shown in the insets in (e),(f).

that using the TR method, the self-bending beams can beobtained, and the beam accurately follows the target trajec-tory. Using simulation and a lens made of elastic materiala similar result is obtained, as shown in Figs. 5(c) and5(d). The experimental tests show a similar pressure-fielddistribution in comparison with theory using the Rayleigh-Sommerfeld integral and simulations using pseudospectralmethods. A lateral shift of the peak pressure location of0.3 mm in the x direction is observed at z = 30 mm andy = 0 mm in the experiments.

Finally, when the aberration layer of the skull phan-tom is included, the corresponding holographic lens alsoreconstructs the target acoustic image with a curved tra-jectory, as shown in Figs. 5(e) and 5(f). A similar lateralshift of the peak pressure location in the experiments, of0.25 mm in the x direction is observed at z = 30 mm andy = 0 mm. The measured pressure field inside the skullphantom agrees with the simulation. Note that the transver-sal size of the curved beam at z = 30 mm is 1.11, 1.07,and 1.19 times the wavelength in water for the theoreticalcalculation, and for the simulations in water and includingthe skull phantom, respectively. Both results demonstratethat using TR methods, self-bending beams following atarget curve can be obtained inside the skull phantom usingacoustic holographic lenses.

V. VOLUMETRIC HOLOGRAMS OVERLAPPINGCNS STRUCTURES

Going further, we design a holographic lens, whichproduces an acoustic image that fits the right human hip-pocampus volume. The holographic surface is placed nearthe occipital-parietal bones to adapt the acoustic image tothe elongated geometry of the hippocampus. However, welocate the lens at the center of the skull symmetry planein order to demonstrate the steering capabilities of thisholographic lens. The lens generation process is based onthe TR method with multiple virtual sources covering thetarget area (see Sec. II for further details). Figure 6 summa-rizes the results for both water and including the aberrationlayer of the skull phantom.

On the one hand, Figs. 6(a)–6(c) show the forward-propagation field distribution of the holographic lensdesigned for water obtained using the theoretical, exper-imental, and simulation results, respectively. First, weobserve a good agreement between experiments, simu-lation, and theory in both axial- [Figs. 6(a)–6(c)] andtransversal-field distributions [Figs. 6(f)–6(h)]. The beamis steered in the correct direction and a broad focal spot isgenerated. The transversal- and axial-field cross sectionsare shown in Figs. 6(d) and 6(e). The diffraction-limited

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FIG. 6. Volumetric hologram results. (a)–(c) Theoretical, experimental, and simulated axial-pressure distribution in water. (d),(e)Transversal- and axial-field cross sections in water. (f)–(h) Simulated, experimental, and theoretical transversal-field distribution inwater. (i),(j) Simulated and experimental axial-pressure distribution including the skull phantom. (k),(l) Corresponding transversal-pressure distribution and (m),(n) transversal and axial cross section, respectively.

image is reconstructed and the field is enhanced mainly atthe target volume. To quantify the performance, we definethe overlapping volume as the overlapping volumes ofthe target region and the region of the acoustic pressurefield under a threshold corresponding to the half of thepeak amplitude. In particular, using this lens we obtain,in water, an overlapping volume of 29.7, 20.1, and 19.0%for the theoretical calculation, simulation, and experiment,respectively.

On the other hand, the field distribution produced byacoustic holographic lenses including the skull phantomis shown in Figs. 6(i)–6(n). The experimental forward-propagation field distribution overlaps a similar volumein comparison with simulation result. Both holographicimages present the same qualitative performance and pro-vide a similar overall covering of the interest zone. Inparticular, an overlapping volume of 21.1 and 23.2% isobtained in simulation and experiment, respectively. Inaddition, both axial- [Fig. 6(i) and 6(j)] and transversal-[Figs. 6(k) and 6(l)] field distributions are similar ofthose produced in water without the skull phantom, show-ing that, first, limited-diffraction holographic volumescan be reconstructed and, second, the aberrations pro-duced by the skull phantom on these complex beamscan be compensated at the source plane by the acous-tic holographic lenses. Finally, the transversal and axialcross sections, shown in Figs. 6(m) and 6(n), show thatthe experimental and simulated acoustic holographic lens

produces a field enhancement that matches the targetdistribution.

Note that the spatial bandwidth of the image is lim-ited by the diffraction limit and the spatial bandwidthof the acoustic holographic lens [21]. In this case, theholographic lens focuses at z ≈ 74.1λ (100 mm), and itslimited aperture is only a = 18.5λ (25 mm). Therefore, thetransversal components of the wave vector are band lim-ited and the performance of the holographic lens at thisdistance is restricted. Using lenses with larger apertureimproves the quality of the holographic acoustic image,and, therefore, the total overlapping volumes.

VI. HOLOGRAM SIMULATION USING AREALISTIC SKULL

It is worth noting here that the impedance of the avail-able 3D printing material used to manufacture the skullphantom is soft compared with the skull bone. In this way,the phase aberrations produced by a real skull are strongerthan the ones observed in the previous experiments. Todemonstrate the focusing performance of the proposedlenses in a realistic situation a set of simulations are per-formed using the acoustical parameters of skull bones. Theparameters are derived using the same CT data and arelisted in Sec. IID.

First, the results of the bifocal lens simulation using arealistic skull are summarized in Fig. 7. First, the sagittal

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FIG. 7. Simulation results for the bifocal holographic lens designed for a realistic skull. (a) Axial cross section of the pressuredistribution at y = 0 mm. (b) Transversal cross section at z = 100 mm. (c) Lateral cross section at z = 100 mm and y = 0 mm. (d)Lateral cross section at z = 100 mm and x = −26 = 0 mm. (e) Axial cross section at x = −26 = 0 mm and y = 0 mm.

cross section of the absolute value of the pressure fieldat y = 0 mm is shown in Fig. 7(a). We can see that thelens focuses at two clear spots, almost at the target dis-tance. The corresponding traversal cross section is shownin Fig. 7(b) measured at z = 100 mm. In fact, good agree-ment is found between the simulations using a realisticskull and the calculations using the Rayleigh-Sommerfeldintegral considering homogeneous water media. Thesetwo focal spots are generated together with small ampli-tude secondary lobes. The amplitude of the side lobes is−8.86 dB below the peak pressure in the theory in water

and −5.16 dB in the simulation including the skull. Toquantify the focusing performance of the lens, we showin Fig. 7(c) the transversal cross section at z = 100 mmand y = 0 mm. The lateral shift of the left focus is −26.3and −26.0 mm for the theoretical prediction and for thesimulation, respectively. A relative error of 1.1% is com-mitted. These small lateral shifts are of the order of theexperimental test shown previously with the 3D-printedphantom. Moreover, the amplitude of the side lobes inthe simulation using a realistic skull are 0.3 times thepeak pressure. These side lobes present higher amplitude

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FIG. 9. Simulation results for the volumetric hologram designed for a realistic skull. (a) Axial cross section of the pressure distribu-tion at y = 0 mm. (b) Transversal cross section at z = 95 mm. (c) Lateral cross section at z = 95 mm and y = 0 mm. (d) Lateral crosssection at z = 95 mm and x = −26 = 0 mm. (e) Axial cross section at x = −26 = 0 mm and y = 0 mm.

in the lateral cross section joining both foci [Fig. 7(c)]than in the lateral cross section measured at x = 0 mm,as shown in Fig. 7(d). Finally, Fig. 7(e) shows the axial-pressure distribution measured at the location of the righthippocampus. The axial peak location of the simulationincluding a realistic skull (z = 99.3 mm) matches the loca-tion of the corresponding peak pressure using the theory inwater (z = 99.8 mm). A relative error of 0.5% is obtained,showing that the aberrations of a real skull can be mitigatedusing holographic lenses even when the target acousticfield presents a complex structure.

Second, the results for the self-bending beam simula-tion inside a realistic skull are shown in Fig. 8. The sagittalcross section is shown in Fig. 8(a), measured at y = 0 mm,together with the location of the target marked in the reddashed line. In this case, the performance of the lens toproduce such a complex beam is reduced as compared withthe previous cases, as can be seen by the presence of sec-ondary lobes. This is mainly caused by the generation ofstrong stationary waves between the skull bone and thelens. However, the peak pressure follows the target tra-jectory and the location of the peak pressure matches thecenter of the curve. A clearer picture is given in Fig. 8(b),which shows the transversal-field distribution measured atz = 35 mm. Here, a sharp focal spot is visible and loca-tion of the peak pressure almost matches the location ofthe target focus. A lateral shift of +0.22 mm, which cor-responds to one numerical grid step in the simulation isfound in the x direction. The corresponding transversal-field distributions are shown in Fig. 8(c) for the x directionand in Fig. 8(d) for the y direction, respectively. Here,

the reference calculations using theoretical methods in ahomogeneous medium (water) are also shown for com-parison. The location of the focal spots observed in bothlateral directions for the simulations including the realis-tic skull are in excellent agreement with the correspondingfocal spots in water. The width of the focal spot obtainedusing both calculation methods also is in agreement. Themain discrepancy is the presence of secondary lobes in thesimulated field, presenting a peak amplitude of 0.36 timesthe pressure at the focus. The amplitude of these lobes is1.7 times larger in direction in which the beam is bent.

Third, a holographic lens is designed with the tar-get of producing a volumetric hologram overlappingwith the right hippocampus volume, and in this caseincluding the acoustic properties of a realistic skull. Theresulting forward-simulated pressure field is shown inFigs. 9(a)–9(e). First, the sagittal cross section of the mag-nitude of the pressure field at y = 0 mm is shown inFig. 9(a). The produced field focuses around the target vol-ume, shown in dashed white lines. The beam is steered inthe direction of the right hippocampus while the transduceraxis remains normal to the skull surface. The transversalcross section at z = 95 mm is shown in Fig. 9(b). While theacoustic field is focused into the target volume, there existsareas not covered by the beam, mainly in the outermostregions away from the transducer source. To quantify thefocusing performance, field cross sections along the cor-responding dashed lines are given in Figs. 9(c)–9(e). Thelateral cross section at z = 95 mm and y = 0 mm is shownin Fig. 9(c). As a comparison, we also plot the correspond-ing cross section of a holographic lens designed to produce

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the same hologram in water using the theoretical Rayleigh-Sommerfeld integration. The simulated beam using therealist skull and the theoretical prediction in water mostlyoverlaps, being the energy of the beam concentrated intothe target volume. Small side lobes with an amplitude 5.3times smaller than the peak pressure are observed in thesimulations including the realistic skull. Note that the cor-responding acoustic intensity of the side lobes is about30 times smaller than the intensity at the focus. The vol-ume of the beam, defined as the total volume of the beamunder a threshold of 0.5 times the peak pressure, roughlyoverlaps with the target volume. The overlapping volumebetween the target and the volumetric acoustic hologramis 29.7% for the theoretical calculation in water and 21.4%for the simulation including the skull. The transversal crosssection along the y axis is given in Fig. 9(d), where excel-lent agreement is found between the two configurations.Finally, the axial cross section along the z axis is shownin Fig. 9(e). In this case, the field presents remarkable sidelobes before and after the target region, but its amplitudeis lower than half of the maximum pressure. Note thisdirection corresponds to the beam axis and the pressuredistribution of the corresponding focused beam presents anelongated shape due to the limited aperture of the source.In this case, a good agreement is also found between theaxial-pressure distribution of the simulation including therealistic skull and the theoretical prediction in water.

VII. CONCLUSIONS

We show that using 3D-printed acoustic holograms itis possible to conform diffraction-limited ultrasonic fieldsof arbitrary shape compensating the aberrations of a thehuman skull. In particular, experimental tests using a 3D-printed skull phantom and numerical simulations using arealistic skull are performed to accurately generate mul-tiple focal holograms, self-bending beams and volumetricholographic fields overlapping a target CNS structure. Theproposed approach using holographic lenses represents astep forward when compared with the existing solutionsusing phase arrays, since it opens alternative venues todevelop reliable and cost-reduced ultrasonic applications.

The quality of the reconstructed acoustic images isrelated to the diffraction limit and the spatial bandwidth ofthe holographic lens, which depends on the spatial aper-ture of the lens, the number of pixels of the lens, andthe frequency of the beam [21]. In this work, we tar-get a human hippocampus using a single holographic lensof only 50-mm aperture and operating at a frequency of1.1 MHz. The reported experimental results inside a skullphantom are in good agreement with theory and simula-tions. Only small shifts, of the order of one wavelength(1.4 mm in water), are found between the target locationand the field produced by the holographic lens. These shiftscan be caused by experimental reasons that include small

positioning error between the lens and the curved surfaceof the phantom and can be corrected using optimizationmethods during lens design. It is worth noting here thatthe phantom used in the experiments presents a smalleracoustic impedance than a real skull. However, full-wavesimulations performed using the density, sound speed, andattenuation values of the skull bone show that these arbi-trary fields can also be produced in a realistic situation. Thegenerated acoustic fields inside the skull are in good agree-ment with those produced in water. This shows that theaberrations produced by the skull can be mitigated by usingholographic lenses even when a complex field is required.

Moreover, using the proposed methodology diffractionis captured exactly as compared with Fraunhofer or angu-lar spectrum methods, leading to a better accuracy of thegenerated acoustic fields. In addition, the holographic lensdesign is based on resonating slabs, which include notonly the refraction over a curved plate, but the resonat-ing waves inside the lens. Finally, it is important to remarkthat the holographic lenses used in this work capture thefull information of the wave field, both phase and ampli-tude are recorded and encoded in the phase-only lens. Thisapproach allows us to reconstruct considerably complexfields.

Phased arrays are efficient but their high cost can beprohibitive to spread out some of the incoming ultra-sonic transcranial therapy treatments. Using phased arraysthe ultrasonic beams can be adjusted in real time andmonitored using MRI, obtaining a precise location of theacoustic focus into the CNS. Nevertheless, the number ofelements of the phased-array systems can be insufficient toproduce a complex volumetric ultrasonic field that matchesa specific CNS structure. The use of holographic lensespresent several advantages to produce complex volumet-ric patterns. First, the cost of a 3D-printed lens is low ascompared with phased-array systems. Second, each pixelin a holographic lens acts as an element of the phasedarray with fixed phase. Due to the high number of pas-sive sources in a holographic lens, more than 4000 forthe small lenses considered here, complex patterns can begenerated. Note that the complexity of the produced acous-tic images can be improved using larger aperture lenses,i.e., increasing the angular spectrum of the recorded holo-graphic information. However, once the lens is designed itsfocal distance and the spatial features of the holographicimage remains fixed and, in principle, it is not possibleto steer the ultrasonic beam in real time with accuracy.For this reason, the technique is specially relevant for thetreatment using a single sonification of structures or in thesonification of large volumes.

The concept shown in this paper opens alternative doorsto optimize and to diseminate incoming therapy treatmentssuch as ultrasound-assisted blood-brain-barrier opening fordrug delivery and neuromodulation, or ultrasonic imag-ing of the central nervous system using low-cost devices.

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Considering the emergence of metamaterials and theirhuge flexibility, we also advance incoming biomedicalapplications of active holographic metasurfaces for thegeneration of complex fields in the central nervous system.

ACKNOWLEDGMENTS

This work is supported by the Spanish Ministry ofEconomy and Innovation (MINECO) through ProjectNo. TEC2016-80976-R. N.J. and S.J. acknowledge finan-cial support from Generalitat Valenciana through GrantsNo. APOSTD/2017/042, No. ACIF/2017/045, and No.GV/2018/11. F.C. acknowledges financial support fromAgència Valenciana de la Innovació through Grant No.INNCON00/18/9 and European Regional DevelopmentFund (Grant No. IDIFEDER/2018/022).

APPENDIX: MEASUREMENT SETUP

The experiments are conducted inside a 1 × 0.75 ×0.5 m3 water tank filled with degassed and distilled waterat 26◦. The ultrasonic transducer is composed by a single-element circular piezoceramic crystal (PZT26, FerropermPiezoceramics, Denmark) mounted in a custom designedstainless-steel housing with aperture 2a = 50 mm asshown in Fig. 10(e). The transducer is driven with a50-cycle sinusoidal pulse burst at a frequency of f =1.112 MHz by a signal generator (14 bits, 100 MS/s,model PXI5412, National Instruments) and amplified bya linear rf amplifier (ENI 1040L, 400 W, 55 dB, ENI,Rochester, NY). The pressure field is measured by a nee-dle hydrophone with a 500-μm active diameter (149.6mV/MPa sensitivity at 1.112 MHz, Model HNR-500,Onda) calibrated from 1 to 20 MHz. The source ampli-tude is set low enough to avoid any nonlinear effects inthe propagation, we measure 1.8 kPa at the focus for thebifocal lens. The hydrophone signals are digitized at asampling rate of 64 MHz by a digitizer (model PXI5620,National Instruments) averaged 100 times to increase thesignal-to-noise ratio.

A 3D micropositioning system (OWIS GmbH) is used tomove the hydrophone in three orthogonal directions withan accuracy of 10 μm. For the bifocal lens experiment,scanning an area for the sagittal cross sections, p(x, z), cov-ers from −40 to 40 mm in the x direction and from 82 to143 mm in the z direction, using a step of 0.5 mm in bothdirections, for the transversal cross-section planes, p(x, y),covers from −40 to 40 mm in the x direction and from −10to 10 mm in the y direction, using a step of 0.3 mm. For theself-bending lens, the scanned area covers from 20 to 50mm in the z direction, from −5 to 5 mm in the x direction,and from −5 to 5 mm in the y direction, using the samespatial steps. Finally, for the volumetric holograms thescanned area covers from 65 to 135 mm in the z direction,from −45 to −5 mm in the x direction, and from −10 to10 mm in the y direction, using the same spatial steps. All

Hologram

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FIG. 10. Experimental setup showing the block diagram andskull phantom inside the water tank with the ultrasonic sourceand the acoustic hologram at the bottom.

the signal generation and acquisition processes are basedon a NI8176 National Instruments PXI-Technology con-troller, which also controlled the micropositioning system.Temperature measurements are performed throughout thewhole process to ensure no temperature changes of 0.5 ◦C.

[1] D. Gabor, A new microscopic principle, Nature 161, 777(1948).

[2] D. Gabor, Microscopy by reconstructed wave-fronts, Proc.R. Soc. Lond. A 197, 454 (1949).

[3] E. N. Leith and J. Upatnieks, Reconstructed wavefronts andcommunication theory, JOSA 52, 1123 (1962).

[4] X. Ni, A. V. Kildishev, and V. M. Shalaev, Metasurfaceholograms for visible light, Nat. Commun. 4, 2807 (2013).

[5] L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen,B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, et al.,Three-dimensional optical holography using a plasmonicmetasurface, Nat. Commun. 4, 2808 (2013).

[6] G. Ma and P. Sheng, Acoustic metamaterials: From localresonances to broad horizons, Sci. Adv. 2, e1501595(2016).

[7] S. A. Cummer, J. Christensen, and A. Alù, Controllingsound with acoustic metamaterials, Nat. Rev. Mater. 1,16001 (2016).

[8] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. T. Chan, andP. Sheng, Locally resonant sonic materials, Science 289,1734 (2000).

[9] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C.Sun, and X. Zhang, Ultrasonic metamaterials with negativemodulus, Nat. Mater. 5, 452 (2006).

[10] M. Yang, G. Ma, Z. Yang, and P. Sheng, Coupled Mem-branes with Doubly Negative Mass Density and BulkModulus, Phys. Rev. Lett. 110, 134301 (2013).

014016-12

HOLOGRAMS TO FOCUS ARBITRARY ULTRASONIC... PHYS. REV. APPLIED 12, 014016 (2019)

[11] Y. Li, B. Liang, Z.-M. Gu, X.-Y. Zou, and J.-C. Cheng,Reflected wavefront manipulation based on ultrathin planaracoustic metasurfaces, Sci. Rep. 3, 2546 (2013).

[12] Y. Xie, W. Wang, H. Chen, A. Konneker, B.-I. Popa, andS. A. Cummer, Wavefront modulation and subwavelengthdiffractive acoustics with an acoustic metasurface, Nat.Commun. 5, 5553 (2014).

[13] N. Jiménez, T. J. Cox, V. Romero-García, and J.-P. Groby,Metadiffusers: Deep-subwavelength sound diffusers, Sci.Rep. 7, 5389 (2017).

[14] N. Jiménez, V. Romero-García, V. Pagneux, and J.-P.Groby, Rainbow-trapping absorbers: Broadband, perfectand asymmetric sound absorption by subwavelength panelsfor transmission problems, Sci. Rep. 7, 13595 (2017).

[15] S. Qi, Y. Li, and B. Assouar, Acoustic Focusing and EnergyConfinement Based on Multilateral Metasurfaces, Phys.Rev. Appl. 7, 054006 (2017).

[16] E. Bok, J. J. Park, H. Choi, C. K. Han, O. B. Wright, and S.H. Lee, Metasurface for Water-to-Air Sound Transmission,Phys. Rev. Lett. 120, 044302 (2018).

[17] Y. Li, X. Jiang, B. Liang, J.-C. Cheng, and L. Zhang,Metascreen-based Acoustic Passive Phased Array, Phys.Rev. Appl. 4, 024003 (2015).

[18] Y. Li and M. B. Assouar, Three-dimensional collimatedself-accelerating beam through acoustic metascreen, Sci.Rep. 5, 17612 (2015).

[19] N. Kaina, F. Lemoult, M. Fink, and G. Lerosey, Nega-tive refractive index and acoustic superlens from multiplescattering in single negative metamaterials, Nature 525, 77(2015).

[20] J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, Experimen-tal demonstration of an acoustic magnifying hyperlens, Nat.Mater. 8, 931 (2009).

[21] K. Melde, A. G. Mark, T. Qiu, and P. Fischer, Hologramsfor acoustics, Nature 537, 518 (2016).

[22] Y. Xie, C. Shen, W. Wang, J. Li, D. Suo, B.-I. Popa, Y. Jing,and S. A. Cummer, Acoustic holographic rendering withtwo-dimensional metamaterial-based passive phased array,Sci. Rep. 6, 35437 (2016).

[23] Y. Zhu, J. Hu, X. Fan, J. Yang, B. Liang, X. Zhu, and J.Cheng, Fine manipulation of sound via lossy metamateri-als with independent and arbitrary reflection amplitude andphase, Nat. Commun. 9, 1632 (2018).

[24] G. Memoli, M. Caleap, M. Asakawa, D. R. Sahoo, B.W. Drinkwater, and S. Subramanian, Metamaterial bricksand quantization of meta-surfaces, Nat. Commun. 8, 14608(2017).

[25] M. D. Brown, B. T. Cox, and B. E. Treeby, Design ofmulti-frequency acoustic kinoforms, Appl. Phys. Lett. 111,244101 (2017).

[26] Y. Hertzberg and G. Navon, Bypassing absorbing objects infocused ultrasound using computer generated holographictechnique, Med. Phys. 38, 6407 (2011).

[27] P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin,and X. Zhang, Generation of acoustic self-bending and bot-tle beams by phase engineering, Nat. Commun. 5, 4316(2014).

[28] A. Marzo, S. A. Seah, B. W. Drinkwater, D. R. Sahoo, B.Long, and S. Subramanian, Holographic acoustic elementsfor manipulation of levitated objects, Nat. Commun. 6,8661 (2015).

[29] M. A. Norasikin, D. Martinez Plasencia, S. Polychronopou-los, G. Memoli, Y. Tokuda, and S. Subramanian, in The 31stAnnual ACM Symposium on User Interface Software andTechnology (ACM, Berlin, Germany, 2018), p. 247.

[30] G. ter Haar and C. Coussios, High intensity focused ultra-sound: physical principles and devices, Int. J. Hyperthermia23, 89 (2007).

[31] A. Bouakaz and J.-M. Escoffre, Therapeutic Ultra-sound: From Biophysics Concepts to Clinical Applications,Advances in Experimental Medicine and Biology Vol. 880(Springer International Publishing, Heidelberg, Germany,2016).

[32] T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out(Academic Press, Oxford, UK, 2004).

[33] P. Gélat, G. Ter Haar, and N. Saffari, A comparison ofmethods for focusing the field of a HIFU array transducerthrough human ribs, Phys. Med. Biol. 59, 3139 (2014).

[34] F. Fry and J. Barger, Acoustical properties of the humanskull, J. Acoust. Soc. Am. 63, 1576 (1978).

[35] J.-L. Thomas and M. A. Fink, Ultrasonic beam focusingthrough tissue inhomogeneities with a time reversal mirror:Application to transskull therapy, IEEE Trans. Ultrason.Ferroelectr. Freq. Control 43, 1122 (1996).

[36] K. Hynynen and F. A. Jolesz, Demonstration of poten-tial noninvasive ultrasound brain therapy through an intactskull, Ultrasound Med. Biol. 24, 275 (1998).

[37] J. Sun and K. Hynynen, Focusing of therapeutic ultrasoundthrough a human skull: A numerical study, J. Acoust. Soc.Am. 104, 1705 (1998).

[38] J.-F. Aubry, M. Tanter, M. Pernot, J.-L. Thomas, andM. Fink, Experimental demonstration of noninvasivetransskull adaptive focusing based on prior computedtomography scans, J. Acoust. Soc. Am. 113, 84 (2003).

[39] M. Tanter, J.-L. Thomas, and M. Fink, Focusing and steer-ing through absorbing and aberrating layers: Applicationto ultrasonic propagation through the skull, J. Acoust. Soc.Am. 103, 2403 (1998).

[40] Y. Hertzberg, A. Volovick, Y. Zur, Y. Medan, S. Vitek, andG. Navon, Ultrasound focusing using magnetic resonanceacoustic radiation force imaging: Application to ultrasoundtranscranial therapy, Med. Phys. 37, 2934 (2010).

[41] F. A. Jolesz, Intraoperative Imaging and Image-GuidedTherapy (Springer Science & Business Media, New York,USA, 2014).

[42] C. Shen, J. Xu, N. X. Fang, and Y. Jing, AnisotropicComplementary Acoustic Metamaterial for Canceling OutAberrating Layers, Phys. Rev. X 4, 041033 (2014).

[43] G. Maimbourg, A. Houdouin, T. Deffieux, M. Tanter, andJ.-F. Aubry, 3D-printed adaptive acoustic lens as a disrup-tive technology for transcranial ultrasound therapy usingsingle-element transducers, Phys. Med. Biol. 63, 025026(2018).

[44] M. Ferri, J. M. Bravo, J. Redondo, and J. V. Sánchez-Pérez,Enhanced numerical method for the design of 3-D-printedholographic acoustic lenses for aberration correction ofsingle-element transcranial focused ultrasound, UltrasoundMed. Biol. 45, 867 (2019).

[45] K. Hynynen, N. McDannold, N. Vykhodtseva, and F. A.Jolesz, Noninvasive MR imaging–guided focal openingof the blood-brain barrier in rabbits, Radiology 220, 640(2001).

014016-13

JIMÉNEZ-GAMBÍN, JIMÉNEZ, BENLLOCH, and CAMARENA PHYS. REV. APPLIED 12, 014016 (2019)

[46] W. J. Tyler, Y. Tufail, M. Finsterwald, M. L. Tauch-mann, E. J. Olson, and C. Majestic, Remote excitation ofneuronal circuits using low-intensity, low-frequency ultra-sound, PloS One 3, e3511 (2008).

[47] U. Schneider, E. Pedroni, and A. Lomax, The calibration ofCT Hounsfield units for radiotherapy treatment planning,Phys Med. Biol. 41, 111 (1996).

[48] T. D. Mast, Empirical relationships between acousticparameters in human soft tissues, Acoust. Res. Lett. Online1, 37 (2000).

[49] J. C. Mazziotta, A. W. Toga, A. Evans, P. Fox, and J. Lan-caster, A probabilistic atlas of the human brain: Theory andrationale for its development, Neuroimage 2, 89 (1995).

[50] P. A. Yushkevich, J. Piven, H. C. Hazlett, R. G. Smith, S.Ho, J. C. Gee, and G. Gerig, User-guided 3D active con-tour segmentation of anatomical structures: Significantlyimproved efficiency and reliability, Neuroimage 31, 1116(2006).

[51] D. T. Blackstock, Fundamentals of Physical Acoustics(John Wiley & Sons, New York, USA, 2000).

[52] B. E. Treeby and B. Cox, Modeling power law absorptionand dispersion for acoustic propagation using the fractionallaplacian, J. Acoust. Soc. Am. 127, 2741 (2010).

[53] B. E. Treeby, J. Jaros, A. P. Rendell, and B. Cox, Modelingnonlinear ultrasound propagation in heterogeneous mediawith power law absorption using ak-space pseudospectralmethod, J. Acoust. Soc. Am. 131, 4324 (2012).

[54] N. Jiménez, F. Camarena, J. Redondo, V. Sánchez-Morcillo, Y. Hou, and E. E. Konofagou, Time-domain sim-ulation of ultrasound propagation in a tissue-like mediumbased on the resolution of the nonlinear acoustic con-stitutive relations, Acta Acust. United Acust. 102, 876(2016).

[55] N. Jiménez, V. Romero-García, V. Pagneux, and J.-P.Groby, Quasiperfect absorption by subwavelength acous-tic panels in transmission using accumulation of resonancesdue to slow sound, Phys. Rev. B 95, 014205 (2017).

[56] P. W. M. Tsang and T.-C. Poon, Novel method for convert-ing digital fresnel hologram to phase-only hologram basedon bidirectional error diffusion, Opt. Express 21, 23680(2013).

[57] R. Lirette and J. Mobley, Focal zone characteristics ofstepped fresnel and axicon acoustic lenses, Proc. Meet.Acoust. 31, 045001 (2017).

[58] M. Gatto, G. Memoli, A. Shaw, N. Sadhoo, P. Gelat, andR. A. Harris, Three-dimensional printing (3DP) of neona-tal head phantom for ultrasound: Thermocouple embeddingand simulation of bone, Med. Eng. Phys. 34, 929 (2012).

[59] J. Robertson, E. Martin, B. Cox, and B. E. Treeby, Sensi-tivity of simulated transcranial ultrasound fields to acousticmedium property maps, Phys. Med. Biol. 62, 2559 (2017).

[60] C. Bai, M. Ji, A. Bouakaz, Y. Zong, and M. Wan,Design and characterization of an acoustically and struc-turally matched 3D-printed model for transcranial ultra-sound imaging, IEEE Trans. Ultrason. Ferroelectr. Freq.Control (2018).

[61] C. R. Hill, J. Bamber, and G. Ter Haar, Physical Princi-ples of Medical Ultrasonics (John Wiley & Sons New York,Chichester, Brisbane, Toronto, 2004).

[62] R. S. Cobbold, Foundations of Biomedical Ultrasound(Oxford University Press, New York, United States, 2006).

[63] H. O’Neil, Theory of focusing radiators, J. Acoust. Soc.Am. 21, 516 (1949).

[64] D.-C. Chen, X.-F. Zhu, Q. Wei, D.-J. Wu, and X.-J. Liu,Broadband acoustic focusing by airy-like beams based onacoustic metasurfaces, J. Appl. Phys. 123, 044503 (2018).

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